Problem 14 - Calculus Single Variable

Question

Solve the given differential equation. $$ x \frac{d y}{d x}-y=3 x^{3} y $$

Step-by-Step Solution

Answer

The solution is \( y = Cx e^{x^3} \), where \( C \) is a constant.

Step 1: Identify the Form

The given differential equation is \( x \frac{d y}{d x} - y = 3x^3y \). This equation can be recognized as a first-order linear differential equation in standard form: \( P(x)y + Q(x) \frac{dy}{dx} = R(x) \). Here, \( P(x) = -3x^3 + 1, \ Q(x) = x, \text{ and } R(x) = 0 \).

Step 2: Rearrange and Isolate Terms

Rewriting the differential equation for clarity, we have: \[ x \frac{dy}{dx} - (1 + 3x^3)y = 0. \] This is another way to express the original problem, making it clearer for the solving technique you'll employ.

Step 3: Recognize the Type

This equation can be solved by using the method of separation of variables. It has variables \( x \) and \( y \) on both sides, and can be manipulated to separate them.

Step 4: Separation of Variables

We rearrange the equation to separate the variables: \[ \frac{1}{y} \frac{dy}{dx} = \frac{1 + 3x^3}{x}. \] Multiply both sides by \( dx \) and \( y \) to get: \[ \frac{1}{y} dy = \frac{1 + 3x^3}{x} dx. \]

Step 5: Integrate Both Sides

Integrate both sides of the equation to find the solutions. Integrating the left side: \[ \int \frac{1}{y} \, dy = \ln |y| + C_1. \]Integrating the right side: \[ \int \left(\frac{1}{x} + 3x^2\right) \, dx = \ln |x| + x^3 + C_2. \]

Step 6: Combine and Simplify

Combining the results from the integration step, we have:\[ \ln |y| = \ln |x| + x^3 + C, \]where \( C = C_2 - C_1 \). Exponentiating both sides to remove the natural logarithm: \[ |y| = e^C x e^{x^3}. \]Let \( e^C = C' \), then \( y = C' x e^{x^3}. \) Note that \( C' \) can be any real number based on the initial conditions.

Step 7: General Solution

The general solution of the given differential equation is:\[ y = C x e^{x^3}, \]where \( C \) is an arbitrary constant determined by initial conditions.

Key Concepts

First-Order Linear Differential Equations

A first-order linear differential equation is one of the simplest forms of differential equations, primarily characterized by the highest derivative being the first derivative. In general, it can be represented in the form

• \( \frac{dy}{dx} + P(x)y = Q(x) \)

where \( P(x) \) and \( Q(x) \) are continuous functions of \( x \).
The equation given, \( x \frac{dy}{dx} - (1 + 3x^3)y = 0 \), fits this category because it is in a format where the highest derivative of \( y \) with respect to \( x \) is of the first order. The task in handling these equations often involves simplifying them to isolate the derivative term and ultimately solve for \( y \).

Separation of Variables

The separation of variables technique is a method used to solve certain kinds of differential equations. This method involves rearranging the equation such that all terms involving one variable are on one side of the equation and all terms involving the other variable are on the opposite side.
For example, the differential equation

• \( \frac{1}{y} \frac{dy}{dx} = \frac{1 + 3x^3}{x} \)

shows the variables \( x \) and \( y \) are separated.
By manipulating the equation, each side can be independently integrated with respect to its variable, leading towards finding the solution.
This strategic move of separation simplifies complex differential terms into more manageable integrals.

Integration Techniques

Integration techniques are paramount when dealing with differential equations, especially when using the separation of variables method. Once separated, each part of the equation can be integrated. For instance:

• The left side: \( \int \frac{1}{y} \, dy = \ln |y| + C_1 \)
• The right side: \( \int \left(\frac{1}{x} + 3x^2\right) \, dx = \ln |x| + x^3 + C_2 \)

In these integrals, we use standard integration formulas.
For example, the integral of \( \frac{1}{y} \, dy \) yields a natural logarithm function, \( \ln |y| \), and similar operations on the right side involving polynomial terms follow straightforward integral rules.
These basic techniques allow us to progress towards simplifying the differential equation into a solvable form.

General Solution of Differential Equations

Once both sides of the separated equation are integrated, the last step involves combining these results to express \( y \) explicitly. This provides the general solution, which features an arbitrary constant representing the family of all possible solutions:

• \( \ln |y| = \ln |x| + x^3 + C \)

Exponentiating both sides removes the logarithm, yielding:

• \( |y| = e^C x e^{x^3} \)

With simplification and substituting \( C' = e^C \), we reach the conclusion:

• \( y = C' x e^{x^3} \)

This formula, \( y = C x e^{x^3} \), where \( C \) is an arbitrary constant, represents the general solution. It captures all potential solutions based on various initial conditions, demonstrating the beauty and flexibility of solving differential equations.